If `(x +y) = sin (x + y) " then " (dy)?(dx)=` ?

Options

A. `-1`
B. 1
C. `(1 - cos (x + y))/(cos^(2) (x + y))`
D. none of these

Explanation

To solve the problem where \( x + y = \sin(x + y) \) and we need to find \( \frac{dy}{dx} \), we will follow these steps:

### Step 1: Differentiate both sides
We start with the equation:
\[ x + y = \sin(x + y) \]
We will differentiate both sides with respect to \( x \).

### Step 2: Apply the chain rule
Using the chain rule, we differentiate the left-hand side:
\[ \frac{d}{dx}(x + y) = \frac{d}{dx}(x) + \frac{d}{dx}(y) = 1 + \frac{dy}{dx} \]
For the right-hand side, we differentiate \( \sin(x + y) \):
\[ \frac{d}{dx}(\sin(x + y)) = \cos(x + y) \cdot \frac{d}{dx}(x + y) = \cos(x + y)(1 + \frac{dy}{dx}) \]

### Step 3: Set the derivatives equal
Now we equate the derivatives from both sides:
\[ 1 + \frac{dy}{dx} = \cos(x + y)(1 + \frac{dy}{dx}) \]

### Step 4: Expand and rearrange
Expanding the right-hand side gives:
\[ 1 + \frac{dy}{dx} = \cos(x + y) + \cos(x + y) \frac{dy}{dx} \]
Now, we rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ 1 - \cos(x + y) = \cos(x + y) \frac{dy}{dx} - \frac{dy}{dx} \]
Factoring out \( \frac{dy}{dx} \) on the right side:
\[ 1 - \cos(x + y) = \frac{dy}{dx}(\cos(x + y) - 1) \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Now we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1 - \cos(x + y)}{\cos(x + y) - 1} \]
This can be simplified further:
\[ \frac{dy}{dx} = -1 \]

### Final Answer:
Thus, the final answer is:
\[ \frac{dy}{dx} = -1 \]